The Stability of Hot Spot Patterns for
Reaction-Diffusion Models of Urban
Crime
Michael J. Ward (UBC)
PIMS Hot Topics: Computational Criminology and UBC (IAM), September 2012
Joint With: Theodore Kolokolonikov (Dalhousie); Simon Tse (UBC); Juncheng Wei
(Chinese U. Hong Kong, UBC)
UBC – p. 1
Modeling Urban Crime I
Multidisciplinary efforts to model patterns of urban crime lead by UCLA
group; A. Bertozzi, P. Brantingham, L. Chayes, M. Short, etc.. (since
2008); field data from LA police; What is best policing strategy?
UC MASC Project: http://paleo.sscnet.ucla.edu/ucmasc.html
UBC – p. 2
Modeling Urban Crime II
Key References:
M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita (2010),
Dissipation and displacement of hotpsots in reaction-diffusion models
of crime, PNAS, 107(9) pp. 3961-3965. Made the cover of PNAS.
M. B. Short, M. R. D’Orsogna, V. B. Pasour, G. E. Tita,
P. J. Brantingham, A. L. Bertozzi and L. B. Chayes (2008), A statistical
model of criminal behavior, M3AS, 18, Suppl. pp. 1249–1267.
M. B. Short, A. L. Bertozzi and P. J. Brantingham (2010), Nonlinear
patterns in urban crime - hotpsots, bifurcations, and suppression,
SIADS, 9(2), pp. 462–483.
Criminal activity concentrates non-uniformly (good versus
bad neighborhoods). Often “hot-spots” of crime are observed. Need to
incorporate near-repeat victimization and elevated risk of re-victimization
in a short time period.
Observations:
UBC – p. 3
An Agent-Based Model I
City is represented by a square lattice. At each lattice
site there is an “attractiveness” A(x, t) and a “number” N (x, t) of criminals.
Criminals exhibit biased random walk and are more likely to move to a
neighboring site with a higher attractiveness.
Agent Based Models:
Criminal Behavior:
Burglarize the house at site x between times t and t + δt
with probablity
pv (x, t) = 1 − e−A(x,t)δt .
If site x is robbed, the burglar is removed from the lattice. After the
attractiveness is updated, burglars are re-introduced at each lattice
site at a rate Γ.
If a burglary at x does not occur, the burglar moves to a neighbouring
site x′ with probability
A(x′ , t)
pm (x ; t, x) = P
.
′′
x′′ ∼x A(x , t)
′
UBC – p. 4
An Agent-Based Model II
Modeling attractiveness:
It has a static and dynamic component:
A(x, t) = A0 + B(x, t) .
Elevated risk of re-victimization in a short time-period with decay rate ω
and with a parameter η ≪ 1 that models how attractiveness is spread to
its neighbours:
"
#
η X
B(x, t + δt) = (1 − η)B(x, t) +
B(x′ , t) (1 − ω δt) + θE(x, t) .
4 ′
x ∼x
Here θ > 0 and E(x, t) is the number of burglary events at site x in a time
interval (t, t + δt). Then, update attractiveness
A(x, t + δt) = A0 + B(x, t + δt)
Remark:
Expected value(E)= N (x, t)pv (x, t).
(stationary hot-spots, moving hot-spots, creation of
new spots, etc..) Agent Based Simulation (M. Short et al:) (Movie)
Numerics (Agent-Based):
UBC – p. 5
The Basic Urban RD Crime Model
In the continuum limit, the resulting dimensionless PDE RD model with no
flux b.c. is (Short et al., M3AS, (2008)):
At = ε2 ∆A − A + P A + α , x ∈ Ω ,
2P
τ Pt = D∇ · ∇P −
∇A − P A + γ − α ,
A
x ∈ Ω.
ε ≪ 1 (results from η ≪ 1)
P (x, t) is criminal density; A(x, t) is “attractiveness” to burglary.
∇A
The chemotactic drift term −2D∇ · P A represents the tendency of
criminals to move towards sites with a higher attractiveness.
Here α is the baseline attractiveness, while γ − α > 0 is the constant
rate of re-introduction of criminals after a burglary.
The spatially homogeneous equilibrium state is
Pe = (γ − α)/γ ,
Ae = γ .
UBC – p. 6
Numerical: Formation of 2-D Hot-Spots
Take P (x, 0) = Pe , A(x, 0) = γ(1 + rand ∗ 0.001) in a square
domain of width 4 with α = 1, γ = 2, ε = 0.08, τ = 1, and D = 1.
2-D Numerics:
A hot-spot pattern emerges on an O(1) time-scale, which then persists.
2
1
0
−1
t=0.0
t=0.6
2
1
0
−1
t=10.0
t=31.5
2
1
0
−1
t=33.4
t=38.2
2
1
0
−1
t=48.6
t=9000.0
−1
0
1
2
−1
0
1
2
UBC – p. 7
Numerical: Formation of 1-D Hot-Spots
Take P (x, 0) = Pe , A(x, 0) = γ(1 − 0.01 cos(6πx)) on an
interval of length 1 with α = 1, γ = 2, ε = 0.02, τ = 1. Plot A(x, t) at
different t. Left: D = 1.0; Right: D = 0.5. Lowering D has increased the
number of stable hot spots. (Explained in more detail later).
1-D Numerics:
2.05
2
1.95
5
0
10
5
0
10
5
0
10
5
0
20
10
0
20
10
0
20
10
0
0
t=0.0
t=13.8
t=25.0
t=30.8
t=6032.3
t=180246.0
t=181178.0
t=250000.0
0.5
2.05
2
1.95
2.5
2
1.5
10
5
0
10
5
0
10
5
0
10
5
0
10
5
0
1
0
t=0.0
t=8.7
t=23.1
t=35.9
t=42.9
t=729.4
t=250000.0
0.5
1
UBC – p. 8
Outline and Perspective I
Mathematical Modeling Remarks:
formulation of stochastic agent-based model based on
observational trends in crime, behavior of criminals, etc...... easy to
incorporate many effects...
Formulation:
numerical realizations of the agent-based model to
observe qualitatively interesting phenomena (stationary hot-spots,
moving hot-spots, creation of new spots, etc..)
Age of Discovery:
Derivation of a “simpler” PDE reaction–diffusion (RD)
system. Usual First Step: Numerical simulations of PDE system,
Turing and weakly nonlinear analysis of patterns.....
Continuum Limit:
existence, regularity, and bifurcation-theoretic
results for the PDE model (Rodriguez, Cantrell, Cosner and
Manasevich).
Rigorous PDE Analysis:
matching model predictions with field observations.
Improving the model. Developing new models (some
game-theoretic)... (Berestycki-Nadal, Pilcher, Short and D’Orsogna).
Model Validation:
UBC – p. 9
Outline and Perspective II
many seemingly “simple-looking” RD systems can exhibit
extremely complex dynamics in the nonlinear regime in different
parameter ranges; i.e witness the Gray-Scott model of chemical physics
(1996–date).
However:
For the RD model of urban crime, use a combination of
formal asymptotics, rigorous analysis, and computation, to obtain
analytical results for stability thresholds, delineating in parameter space
where different solution behaviors occur, etc...
Our Approach:
Analyze the existence and stability of localized patterns of
criminal activity for this model in the limit ε → 0. We also consider
extensions of the basic model to include the effect of “police”. These
patterns are “far-from-equilibrium” (Y. Nishiura..), and not amenable to
standard Turing stability analysis.
Specific Goal:
Particle-like solutions to PDE’s; vortices, skyrmions, hot-spots, ...
UBC – p. 10
Turing-Stability Analysis
The uniform state Ae , Pe on R1 is unstable for ε → 0 when γ > 3α/2. With
an eimx+λt perturbation, the Turing instability band is
D−1/2 γ(2γ − 3α)−1/2 ∼ mlower < m < mupper ∼ ε−1 γ −1/2 (2γ − 3α)1/2 .
For ε → 0, the maximum growth rate is λmax ∼ O(1), with the most
unstable mode
1/4
−1/2 −1/4 −1/2
2
.
mmax ∼ ε
D
γ
(γ − α)(3γ + 2τ (2γ − 3α)
For perturbations of the uniform state the preferred pattern has a
characteristic half-length lturing ∼ π/mmax , where
Remark 1:
lturing ∼ ε
1/2
D
1/4 1/2
γ
2
(γ − α)(3γ + 2τ (2γ − 3α))
Notice that lturing = O(1) when D = O(ε−2 ).
−1/4
π.
Turing and weakly nonlinear analysis in 1-D and 2-D given in
Short et al. (SIADS 2010), together with full numerical computations.
Remark 2:
UBC – p. 11
Global Bifurcation Diagram in 1-D
A(0) versus γ on 0 < x < 1 for α = 1, ε = 0.05, D = 2.
Notice that a localized hot-spot occurs when A(0) is large.
Bifurcation Diagram:
9
8
6
8
7
6
4
A(0)
6
2
2
1.8
5
0
1.6
0.5
1
1.4
4
3
2.5
2
1.5
3
2
0
0.5
1
0
0.5
1.6
1.56
1.48
1.52
0
11
1.1
1.2
1.3
1.4
1
1.5
0.5
1.6
1
1.7
γ
Caption: Ae = γ is the solid line with shallow slope; The hot-spot
asymptotics (to be derived) is the dotted line A(0) ∼ 2(γ − α)/(επ).
Subcritical Turing occurs at γ = 3α/2 + O(ε) ≈ 1.5, with weakly nonlinear
asymptotics (dashed parabolic curve). Inserts plot A(x) versus x.
UBC – p. 12
Basic RD Crime Model: Qualitative I
There are two key parameter regimes:
Regime 1:
D ≫ 1 and D = O(1)
D≫1
Localized patterns in the form of pulses or spikes are readily
constructed using singular perturbation techniques for the regime
O(1) ≪ D ≤ O(ε−2 ) in 1-D and O(1) ≪ D ≤ O(ε−4 ) in 2-D.
The stability threshold for these patterns occurs when D = O(ε−2 ) in
1-D and D = O(ε−4 ) in 2-D.
The stability theory is based primarily on an exactly solvable nonlocal
eigenvalue problem.
A further stability threshold in D with respect to instabilities developing
over a long time scale t = O(ε−2 ) must also be calculated.
The stability threshold in terms of D determines the mininum
spacing between localized elevated regions of criminal activity that allows
for a stable pattern, i.e. If Dcrit is large, stable hot-spots are closely
spaced.
Implication:
UBC – p. 13
Localized Hot-Spot Patterns in 1-D: History
A rather extensive literature on the stability of pulses for two-component
RD systems without gradient terms. Prototypical is
vt = ε2 vxx − v + v 2 /u ,
τ ut = Duxx − u + ε−1 v 2 ,
(GM model).
NLEP stability theory (1999-date) (Iron, Kolokolnikov, Ward,
Wei, Winter; Doelman, Gardner, Kaper, Van der Ploeg,..).
History:
Let w′′ − w + w2 = 0 be the homoclinic. To study the stability on an
O(1) time-scale need to analyze the spectrum of the NLEP
R∞
wΦ dy
2 −∞
= λΦ ; Φ → 0 as |y| → ∞.
Φyy − Φ + 2wΦ − χ(τ λ)w R ∞ 2
w
dy
−∞
If D > Dc , K > 1, and τ < τc , then there is a sign-fluctuating
instability of the spike amplitudes due to a positive real eigenvalue;
this is a competition instability (Iron-Ward-Wei (Physica D, 2001)).
If D < Dc , K > 1, but τ > τc , then there is a synchronous oscillatory
instability of the spike amplitudes due to a Hopf bifurcation.
(Ward-Wei, (J. Nonl. Sci, 2003)).
UBC – p. 14
Regime 1: Hot-Spot Equilibria in 1-D: I
Basic Cell Problem:
Since Px −
2P
A
Consider WLOG the interval |x| < l.
Ax = (P/A2 )x A2 , we let V = P/A2 . Then, on |x| < l
At = ε2 Axx − A + V A3 + α ,
2
2
τ A V t = D A V x x − V A3 + γ − α .
For D = O(ε−2 ), the correct scaling is
V = ε2 v ,
D = D0 /ε2 .
Therefore, our re-scaled RD system on the basic cell |x| < l is
At = ε2 Axx − A + ε2 vA3 + α ;
Ax (±l, t) = 0 ,
2
2
2
ε τ A v t = D0 A vx x − ε2 vA3 + γ − α ;
vx (±l, t) = 0 .
A = O(ε−1 ) in the core of a hot-spot while A = O(1) away from
the core. We obtain that v = O(1) globally.
Remark:
UBC – p. 15
Regime 1: Hot-Spot Equilibria in 1-D: II
From a matched asymptotic analysis:
Let ε → 0 and O(1) ≪ D ≤ O(ε−2 ) with D0 ≡ ε2 D. Then,
for a one-hot-spot solution centered at x = 0 on |x| ≤ l, the leading-order
uniform asymptotics for A and P are
1
2l(γ − α)
2
A∼ √
− α w(x/ε) + α ,
P ∼ [w(x/ε)] .
πε
2
√
Here w(y) = 2sech (y) is the homoclinic of w′′ − w + w3 = 0. The inner
and outer approximations for v are
Principal Result:
ζ
2
2
v ∼ v0 + εv1 , |x| = O(ε) ; v ∼
(l − |x|) − l + v0 , O(ε) < |x| < l ,
2
2
2 −1
2
2
Here ζ ≡ (α − γ)/(D0 α ) < 0 and v0 = π 2l (γ − α)
.
Remarks:
A(0) ∼ 2(γ − α)/(επ), as plotted previously on bifurcation diagram.
For a symmetric K-hot-spot pattern with spots of equal height on a
domain of length S, simply set l = S/2K and use gluing.
UBC – p. 16
Regime 1: Hot-Spot Equilibria in 1-D: III
Comparison with Full Numerics:
with D0 = 1, γ = 1, ε = 0.05, and α = 2:
14
12
10
A
8
6
4
2
0
ε
0.1
0.05
0.025
0.0125
Remark:
A(0) (num)
6.281
12.805
25.628
51.145
0.2
0.4
A(0) (asy1)
6.366
12.732
25.465
50.930
x
0.6
0.8
v(0) (num)
3.5844
4.1474
4.4993
4.7039
1
v(0) (asy1)
4.935
4.935
4.935
4.935
v(0) (asy2)
2.961
3.948
4.441
4.688
A one-term approximation for A(0) is accurate even for ε = 0.1.
UBC – p. 17
NLEP Stability Analysis: I
Let Ae , ve be one-spike solution on the basic cell |x| < l. We introduce
A = Ae + φeλt ,
v = ve + εψeλt .
The singularly perturbed eigenvalue problem is
D0
ε2 φxx − φ + 3ε2 ve A2e φ + ε3 A3e ψ = λφ ,
2
2 2
3
3
2
2
εAe ψx + 2Ae vex φ x − 3ε Ae ve φ − ε ψAe = λτ ε εAe ψ + 2Ae ve φ .
For z complex, we impose the Floquet boundary conditions
φ(l) = zφ(−l) ,
φ′ (l) = zφ′ (−l) ,
ψ(l) = zψ(−l) ,
ψ ′ (l) = zψ ′ (−l) ,
To obtain the spectrum of a K-spike pattern on a domain of length 2Kl
subject to periodic boundary conditions we set z K = 1, so that
zj = e2πij/K ,
j = 0, . . . , K − 1 .
An NLEP is derived for the periodic b.c. problem by using
asymptotics to determine jump conditions for ψ across x = 0. Then, the
Neumann spectra is extracted from Periodic spectra.
Remark:
UBC – p. 18
NLEP Stability Analysis: II
An asymptotic analysis shows that φ ∼ Φ(y) with y = ε−1 x, where Φ(y) on
−∞ < y < ∞ satisfies
−3/2
L0 Φ ≡ Φ′′ − Φ + 3w2 Φ = −v0
w3 ψ(0) + λΦ ;
Φ → 0 as |y| → ∞.
Then, for τ = O(1), ψ(0) is determined from
ψxx = 0 ,
0 < |x| ≤ l ;
ψ(l) = zψ(−l) ,
ψ ′ (l) = zψ ′ (−l) ,
subject to ψ(0+ ) = ψ(0− ) ≡ ψ(0) and the jump condition across x = 0:
2
a0 ≡ D0 α ,
Remark:
a0 [ψx ]0 + a1 ψ(0) = a2 ;
Z ∞
Z
−3/2
a1 = −v0
w3 dy
a2 = 3
−∞
∞
w2 Φ dy
−∞
ψ(0) involves one nonlocal term.
UBC – p. 19
NLEP Stability Analysis: III
Consider a K > 1 equilibrium hot-spots on an interval of
length S with no-flux conditions. For ε → 0, τ = O(1), and with D0 = ε2 D,
the stability of this solution with respect to the “large” eigenvalues
λ = O(1) of the linearization is determined by the spectrum of the NLEP
R∞ 2
w Φ dy
3 −∞
= λΦ ;
Φ → 0 as |y| → ∞ ,
L0 Φ − χj w R ∞ 3
w dy
−∞
"
#−1
4
D0 α 2 π 2 K 4 2
χj = 3 1 +
, j = 0, . . . , K − 1.
(1 − cos (πj/K))
3
4(γ − α)
S
Principal Result:
For K = 1 we have χ0 = 3. Here L0 (local operator) is
L0 Φ ≡ Φ′′ − Φ + 3w2 Φ
In contrast to the NLEP’s for the GM and GS models, the discrete
spectrum for this NLEP is explicitly available.
Remark:
UBC – p. 20
NLEP Stability Analysis: IV
Lemma:
Let c be real, and consider the NLEP on −∞ < y < ∞:
Z ∞
w2 Φ dy = λΦ ;
Φ → 0 as |y| → ∞ ,
L0 Φ − cw3
−∞
R∞
corresponding to −∞ w2 Φ dy 6= 0. On the range Re(λ) > −1, there is a
unique discrete eigenvalue given by
Z ∞
λ=3−c
w5 dy .
−∞
Thus, λ is real and λ < 0 when c > 3/ w5 dy.
R
2
2
We use the
key
identity
L
w
=
3w
and Green’s theorem
0
R
∞
w2 L0 Φ − ΦL0 w2 dy = 0 with L0 Φ = cw3 −∞ w2 Φ dy + λΦ. Thus,
Z ∞
Z ∞
w2 Φ dy = 0 ,
w5 dy
λ−3+c
Idea of Proof:
R∞
−∞
−∞
−∞
which yields the result.
UBC – p. 21
NLEP Stability Analysis: V
By applying this Lemma to our NLEP, we get:
On a domain of length S with Neumann b.c., for τ = O(1)
and D0 = ε2 D a one-hot-spot solution is stable on an O(1) time-scale
L
∀D0 > 0. For K > 1 it is stable on an O(1) time-scale iff D0 < D0K
, where
Principal Result:
4
L
D0K
2(γ − α)3 (S/(2K))
.
≡ 2 2
α π [1 + cos (π/K)]
In terms of the original diffusivity D, given by D = ε−2 D0 , the stability
L
when K > 1.
threshold is DKL = ε−2 D0K
There are “small” o(1) eigenvalues in the linearization
that are difficult to asymptotically calculate directly. The threshold in D for
this critical spectrum is obtained indirectly by determining the value DKS of
D for which a asymmetric K-hot-spot equilibrium branch bifurcates from a
symmetric K-hot-spot branch. We readily calculate that
4
3
(γ − α)
S
S
= 2 2 2
.
DK
ε π α
2K
Small Eigenvalues:
UBC – p. 22
NLEP Stability Analysis: VI
Summary:
The small and large (NLEP) eigenvalue stability thresholds are
DKS ∼
S
2K
4
3
(γ − α)
,
2
2
2
ε π α
DKL = DKS
2
1 + cos (π/K)
> DKS .
Thus, we have stability wrt both classes of eigenvalues when
S
S
L
D < DK
; a weak translational instability when DK
< D < DK
; a fast O(1)
L
.
time-scale when D > DK
Remark 1:
Remark 2 (KEY):
For stability, we need the inter-hot-spot spacing l to satisfy
l > lc , where
lc ∼
√
πD1/4 ε1/2 α1/2 (γ − α)−3/4
Since lc and lturing are both O(1) when D = O(ε−2 ), the
maximum number of stable hot-spots corresponds (roughly) to the most
unstable Turing mode.
Remark 3 (KEY):
UBC – p. 23
Numerical Validation I
Let ε = 0.07, α = 1, γ = 2, S = 4, τ = 1, and K = 2, so that
DKS ≈ 20.67 and DKL ≈ 41.33. The results below confirm the stability theory.
Full Numerics:
Left: D = 15
Middle: D = 30
Right: D = 50
UBC – p. 24
Qualitative Implications: Stability Analysis
On an interval of length S, the stability properties of a K-hot-spot
equilibrium pattern can be phrased in terms of the maximum number of
hot-spots:
Unstable wrt a competition instability developing on an O(1) time scale
if K > Kc+ , where Kc+ > 0 is the unique root of
1/4
2
(γ − α)3/4
S
1/4
√
.
K (1 + cos (π/K))
=
2
D
πεα
Stable with respect to slow translational instabilities developing on an
O(ε−2 ) time-scale if K < Kc− < Kc+ , where
3/4
(γ
−
α)
S
D−1/4 √
Kc− =
.
2
πεα
stability when K < Kc− ; stability wrt O(1) time-scale
instabilities but unstable wrt slow translation instabilities when
Kc− < K < Kc+ ; a fast O(1) time-scale instability when K > Kc+ .
Summary:
UBC – p. 25
Numerical Validation II
Take α = 1, γ = 2, and ε = 0.02.
Left (D = 1): Kc+ ≈ 2.27 and Kc− ≈ 1.995. Thus, 3-spots are NLEP
unstable (O(1) time-scale instability), and 2-spots (unstable on very long
time-interval).
Right (D = 0.5): Kc+ ≈ 2.61 and Kc− ≈ 2.37. Thus, 3-spots are NLEP
unstable, but 2-spots are stable wrt NLEP and translations.
1-D Numerics (Revisited):
2.05
2
1.95
5
0
10
5
0
10
5
0
10
5
0
20
10
0
20
10
0
20
10
0
0
t=0.0
t=13.8
t=25.0
t=30.8
t=6032.3
t=180246.0
t=181178.0
t=250000.0
0.5
2.05
2
1.95
2.5
2
1.5
10
5
0
10
5
0
10
5
0
10
5
0
10
5
0
1
0
t=0.0
t=8.7
t=23.1
t=35.9
t=42.9
t=729.4
t=250000.0
0.5
1
UBC – p. 26
Quasi-Equilibria and NLEP Stability in 2-D
In a 2-D domain Ω with area |Ω|.
Principal Result: For ε → 0 and τ = O(1), a symmetric K-hot-spot
quasi-steady-state solution for D = ε−4 D0 /σ with σ = −1/ log ε, is
characterized near the j-th hot-spot by
w(ρ)
A ∼ 2√ ,
ε v0
2
P ∼ [w(ρ)] ,
with ρ = ε−1 |x − xj |. Here w(ρ) is the radially symmetric ground-state of
∆ρ w − w + w3 = 0. In addition,
Z ∞
4π 2 b2 K 2
2
v0 ≡
b
=
ρw
dρ .
|Ω|2 (γ − α)2
0
this quasi-steady-state for K > 1 is stable wrt O(1) instabilities of the
associated NLEP iff
−4 3
ε
|Ω|3 (γ − α)
L
L
D<
D0K , where D0K ≡
2 .
R
3
2
3
σ
4πα K
2 w dy
R
For K = 1 a single hot-spot is stable for all D0 independent of ε.
UBC – p. 27
Basic RD Crime Model: Qualitative II
Regime 2:
D = O(1)
At = ε2 ∆A − A + P A + α , x ∈ Ω ,
2P
τ Pt = D∇ · ∇P −
∇A − P A + γ − α ,
A
x ∈ Ω.
Localized hot-spots still exist, but
In 1-D, localized regions of criminal activity can be nucleated in the
region between neighboring hot-spots when the inter hot-spot spacing
exceeds some threshold.
Leads to the “spontaneous” creation of new hot-spots, i.e. new regions
of elevated criminal activity.
This is called peak-insertion in R-D theory, and it arises from a
saddle-nose bifurcation point in terms of D.
In 1-D the hot-spot dynamics is repulsive, and a reduction to
finite-dimensional dynamics can be done.
UBC – p. 28
Peak-Insertion or Nucleation: D = O(1)
ε = 0.02, α = 1, γ = 2, τ = 1, with x ∈ (0, 2). when D is slowly
decreasing in time as D = 1/(1 + 0.01t). Plot of P (x). Peak inserton
occurs whenever D is quartered.
Numerics:
4
2
D=0.866
2
0
2
D=0.835
1
0
2
1
2
0
1
0
2
D=0.797
4
D=0.808
1
2
0
2
0
2
D=0.753
1
2
0
D=0.269
1
1
1
0
0
0
0
2
1
2
2
D=0.246
1
0
0
1
2
2
D=0.060
1
0
1
2
0
0
1
2
D=0.058
1
2
0
1
2
D=0.261
0
2
1
0
0
2
1
0
D=0.804
1
2
D=0.040
1
0
1
2
0
0
1
2
UBC – p. 29
Peak-Insertion Behavior: Global AUTO
Global Bifurcation Diagram (AUTO): A Two-Boundary and One-Interior Spike
Solution for γ = 2, α = 1, ε = 0.02, on a domain of length 2. Top: |A|2
versus D: Bottom: P (x).
75.0
1
72.5
2
70.0
L2-NORM
67.5
3
65.0
62.5
5
57.5
0.25
6
7
0.30
0.35
1.75
PAR(2)
0.40
1.75
1
1.00
3
0.50
1.75
5
1.50
1.50
1.25
1.25
1.25
1.00
1.00
1.00
U(3)
1.25
1.75
U(3)
1.50
U(3)
1.50
0.45
6
U(3)
60.0
4
0.75
0.75
0.75
0.75
0.50
0.50
0.50
0.50
0.25
0.25
0.25
0.25
0.000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t
0.000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t
0.000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t
0.000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t
Labels:
1
3
5
6
UBC – p. 30
Analysis of Peak-Insertion Behavior: I
Consider WLOG a one-spike pattern centered at the
2
2
midpoint of the interval |x| < l. Since Px − 2P
A Ax = (P/A )x A , we define
Basic Cell Problem:
V = P/A2 .
Then, on |x| < l, the equilibrium problem is
ε2 Axx − A + V A3 + α = 0 ,
2
D A V x x − V A3 + γ − α = 0 ,
Goal:
Ax (±l) = 0,
Vx (±l) = 0.
√
Use ε ≪ 1 asymptotics to determine A(l) versus l/ D.
2.00
2.00
1.75
1.75
1.75
1.50
1.50
1.50
1.25
1.25
1.25
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.000.0
0.2
0.3
0.4
0.5
t
0.6
0.7
0.8
0.9
1.0
0.000.0
1.00
0.75
8
0.50
9
0.25
10
0.1
U(3)
2.00
U(3)
U(3)
As D decreases, peak insertion for P (x) at the boundary occurs:
0.25
0.1
0.2
0.3
0.4
0.5
t
0.6
0.7
0.8
0.9
1.0
0.000.0
0.1
0.2
0.3
0.4
0.5
t
0.6
0.7
0.8
0.9
1.0
UBC – p. 31
Analysis of Peak-Insertion Behavior: II
Inner Region:
We set y = x/ε, and expand
A ∼ ε−1 A0 + A1 + · · · ,
We obtain that
′′
p
A0 = w/ ṽ0 ,
w=
V ∼ ε2 ṽ0 + · · ·
√
2sech(y) ,
where w − w + w3 = 0. Here ṽ0 is to be found, and A1 → α as y → ±∞.
WLOG consider 0+ < x ≤ l. Key: The leading order outer
problem is nonlinear. With, A ∼ a0 and V ∼ v0 , then
2
3
D a0 v0x x = v0 a30 − (γ − α) ,
v0 a0 = a0 − α ,
Outer Region:
which leads to
D (f (a0 )a0x )x = a0 − γ ,
0 < x ≤ l;
a0 (0+ ) = α ,
a0x (l) = 0 ,
where
f (a0 ) ≡ a−2
0 (3α − 2a0 ) .
UBC – p. 32
Analysis of Peak-Insertion Behavior: III
We have f (a0 ) > 0 and a0x > 0 when a0 < 3α/2 < γ. Note: for
γ > 3α/2 the spatially homogeneous steady-state is Turing unstable.
Remark 1:
Remark 2:
For the existence of a solution a0 we require a0 (l) ≡ µ ≤ 3α/2.
Upon integrating
√ the BVP for a0 , we obtain an implicit relation for µ ≡ a0 (l)
in terms of l/ D. Namely,
r
Z µ
p
p
2
2
1
l = χ(µ) ≡
G(α; µ) + 2
G(η; µ) dη ,
2
D
γ−α
α (η − γ)
where
G(η; µ) ≡
Z
µ
η
1
µ
1
f (s)(γ−s) ds = 2(µ−η)−(2γ+3α) log
+3αγ
−
.
η
η µ
In addition, the unknown constant ṽ0 for the inner solution is
π2
−1
[G(α; µ)] .
ṽ0 =
4D
UBC – p. 33
Analysis of Peak-Insertion Behavior: IV
Key Monotonicity Properties:
Gµ (η; µ) > 0 and χ′ (µ) > 0.
√
Upshot: As µ = a0 (l) increases on α < µ < 3α/2, then l/ D increases.
Main Result:
For l fixed, we require D > Dmin , where
2
Dmin = 2l2 / [χ(3α/2)] .
+
, then a0x (l) → +∞, and we predict peak insertion.
As D → Dmin
Corollary:
For D fixed, we require l < lmax , where
p
lmax = χ (3α/2) D/2 .
−
, then a0x (l) → +∞, and we predict peak insertion.
As l → lmax
The analytical theory gives Dmin ≈ 0.445 for l = 1/2. Full
numerics with AUTO gives Dmin ≈ 0.27 for ε = 0.02 and Dmin ≈ 0.44 for
ε = 0.0027.
Comparison:
Remark
The error in the outer approximation is O(−ε log ε).
UBC – p. 34
Analysis of Peak-Insertion Behavior: V
perform a local analysis near x = l when D ≈ Dmin in order to
describe the nucleation of the new peak.
Goal:
Can we analytically uncover solution multiplicity arising from a
saddle-node bifurcation near Dmin ?
Q1:
If so, is there some normal form-type equation that can be
rigorously analyzed describing the local behavior of solutions near the
saddle-nose transition?
Q2:
Local Analysis:
Define the constants Ac , Vc , β, σ, and ζ by
3α
(γ − 3α/2)
Ac ≡
,
Vc ≡ F(Ac ) ,
β≡
> 0,
2
2Dmin A2c
1/6
2/3
−2
−2
2
, ζ = Ac β
,
σ=
A2c F ′′ (Ac )β
A2c F ′′ (Ac )β
′′
where F(A) ≡ (A − α)/A3 with F (Ac ) < 0.
UBC – p. 35
Analysis of Peak-Insertion Behavior: VI
Then, near the endpoint x = l, we obtain the local
approximation
∗
2
2/3
4/3
2
A ∼ Ac − ε ζ U (y) , V ∼ Vc − ε βσ A + y ,
Main Result:
where y = (l − x)/(ε2/3 σ). The function U (y) on y ≥ 0 satisfies the normal
form non-autonomous ODE
2
∗
2
Uyy = U − A − y ,
Remark:
y ≥ 0;
Uy (0) = 0 ;
′
U → +1 , y → +∞ .
If U (0) < 0, then A > Ac = 3α/2.
Main Result: For A∗
≫ 1, there are two solutions U ± (y) with U ′ > 0 for
y > 0 given asymptotically by
p
√
+
+
∗
2
U ∼ A +y ,
U (0) ∼ A∗ ,
!!
√
p
√
A∗ y
−
−
2
∗
2
√
,
U (0) ∼ −2 A∗ .
U ∼ A + y 1 − 3 sech
2
Rigorous: these solutions are connected via a saddle-node bifurcation.
UBC – p. 36
Analysis of Peak-Insertion Behavior: VII
The same normal form ODE was derived for the analysis of
self-replicating mesa patterns in reaction-diffusion systems: see KWW,
Physica D 236(2), (2007), pp. 104-122.
Remark 2:
Plot of A∗ versus s
≡ U (0):
A
∗
8
6
5
5
5
–5
0
–5
–5
∗
s = −5, A ∼
–6
–4
25
4
4
2.5
5
0
–1
5
∗
2
s = −0.615, A = −1.466
–2
0
–5
0
5
s = 2.5, A∗ ∼
2
25
4
s4
–2
UBC – p. 37
Finite-Dimensional Dynamics: D = O(1): I
On the basic cell, |x| ≤ l, one can construct a quasi-equilibrium one-spike
solution centered at x = x0 , where x0 = x0 (ε2 t) moves slowly in time.
Provided that no peak-insertion effects occur, then for ε → 0
the dynamics on the slow time-scale σ = ε2 t is characterized by
p
−1
A ∼ ε w (y) / ṽ0 , y = ε−1 (x − x0 (σ)) ,
3
dx0
+
−
∼
F(x0 ) ,
F(x0 ) ≡ a0x (x0 ) + a0x (x0 ) .
dσ
8α
Main Result:
Here a0 (x) is the solution to the multi-point BVP
D [f (a0 )a0x ]x = a0 − γ ,
0 < |x| < l ;
a0x (±l) = 0 ,
a0 (0) = α ,
where f (a0 ) ≡ a−2
0 (3α − 2a0 ).
The derivation of this is delicate in that one must resolve a
corner layer or knee region for V that allows for matching between the
inner and outer approximations for V .
Remark 1:
UBC – p. 38
Finite-Dimensional Dynamics D = O(1): II
From an integration of the BVP for a0 , one gets explicit dynamics:
h
i
p
p
3 2
dx0
=
G(α; µr ) − G(α; µl ) ,
dσ
8 D
where µr ≡ a0 (l) and µl ≡ a0 (−l) are determined implicitly by
r
r
2
2
(l − x0 ) = χ(µr ) ,
(l + x0 ) = χ(µl ) .
D
D
The dynamics is repulsive: If x0 (0) > 0, then x0 → 0 as
t → ∞. By using reflection through the Neumann B.C., two adjacent
hot-spots will repel.
Qualitative I:
Since the hot-spot dynamics is repulsive, then dynamic
peak-insertion events due to large inter-hot-spot separations are unlikely.
Qualitative II:
Peak insertion events in the presence of mutually attracting
localized pulses, leads to spatial temporal chaos in a Keller-Segel model
with logistic growth in 1-D (Painter and Hillen, Physica D 2011).
Remark:
UBC – p. 39
The Effect of Police: 3-Component Systems
In 1-D, an RD system incorporating police is U = U (x, t):
At = ε2 ∆A − A + P A + α , x ∈ Ω ,
2P
Pt = D∇ · ∇P −
∇A − P A + γ − α − f , x ∈ Ω ,
A
qU
∇A , x ∈ Ω ,
τu Ut = D∇ · ∇U −
A
R
with ∂n (A, P, U ) = 0 on ∂Ω. Police are conserved; U0 = Ω U dx > 0 for all t.
Police Model I:
f = U (simple interaction) L. Ricketson (UCLA).
Police Model II:
f = U P (standard “predator-prey” type interaction)
Remark:
The police drift velocity is V =
d
dx
ln(Aq ).
If q = 2, police drift exhibits mimicry (L. Ricketson, UCLA) Referred to
as “Cops on the Dots” (Jones, Brantingham, L. Chayes, M3As, 2011).
Can be derived from an agent-based model.
If q > 2, police focus more on attractive sites than do criminals.
If 0 < q < 2, the police are less focused, and more “diffusive”.
UBC – p. 40
The Effect of Police: Qualitative
Optimal Police Strategy: For a given U0 find the optimal q
(parameter in police drift velocity) that minimizes the NLEP stability
threshold of D with D = D0 /ε2 . In this way, we maximize over q the
distance between stable localized hot-spots.
Question I:
Investigate this question for both Police models I and II.
Is the optimal strategy the same for both models?
For τu sufficiently large on the regime D = O(ε−2 ),
corresponding to police diffusivity D/τu , can a two-hot spot solution
undergo a Hopf bifurcation leading to asynchronous temporal oscillations
in the hot-spot amplitudes?
Question II:
If τu > 1, then police ‘diffuse” more slowly than do criminals.
Typically, only synchronous oscillatory instabilities of the spike
amplitudes occur in RD systems (GM, Gray-Scott, etc..)
Open: For D = O(1) can police prevent or limit the nucleation
of new hot-spots of criminal activity arising from peak-insertion?
Question III:
UBC – p. 41
Police Model I: Simple-Interaction Model
On the basic cell |x| > l, and with q > 1, the
leading order asymptotics for A, P , and U , in the hot-spot region near
x = 0 is
Principal Result (Equilibrium):
U0
1
2
q
√
w (x/ε) , P ∼ [w (x/ε)] , U ∼
A∼
[w (x/ε)] .
εb
ε v0
√
R q
Here b ≡ w dy, and w = w(y) = 2sech (y) is the homoclinic of
w′′ − w + w3 = 0. The amplitude of the hot-spot is determined by v0 , where
Z
1
√
w3 dy = 2l(γ − α) − U0 .
v0
Remark I:
A hot-spot solution ceases to exist if the total number of police
satisfies
U0 ≥ U0c ≡ 2l(γ − α)
For U0 < U0c , for a K-spot equilibrium on a domain of length S,
let l = S/(2K) and replace U0 → U0 /K. Then, use a glueing technique to
construct the multi-pulse pattern.
Remark II:
UBC – p. 42
Police Model I: Stability I
For q > 1, a Floquet-based analysis leads to an NLEP with two nonlocal
terms:
R 2
Z
w Φ dy
L0 Φ − 3χ0j w3 R 3
− χ1j w3 wq−1 Φ dy = λΦ ,
w dy
where for j = 1, . . . , K − 1,
−1
Z
3/2
,
χ0j ≡ 1 + v0 Dj2 / w3 dy
Cq (λ)
χ1j ≡ R 3
χ0j .
w dy
Here v0 , Djq , and Cq (λ) are defined by
R 3
w dy
2K
U0
πj
S
, Djq ≡ D0 αq
1 − cos
= (γ − α) −
√
v0
K
K
S
k
R q !
√
U
w
dy
qκp Djq
0 v0
q−3
R
Cq (λ) ≡
, τ̂ ≡ ε τu
.
, κp ≡
q dy
q/2
Djq + τ̂ λ
K
w
v
0
For q = 3 it is an exactly solvable NLEP.
By using L0 (w2 ) = 3w2 , the NLEP can be transformed into one
with a single nonlocal term.
Remark 1:
Remark 2:
UBC – p. 43
Police Model I: Stability IV
For q > 1, the NLEP governing the stability of a K-hot-spot
pattern is for j = 1, . . . , K − 1:
R q−1
Φ dy
1
3 w
R
L0 Φ −
w
= λΦ ,
Cj (λ)
wq dy
9χ0j
1
R
1−
.
Cj (λ) =
2(3 − λ)
χ1j wq dy
Main Result:
The discrete eigenvalues are the roots of gj (λ) = 0, where
R q−1
−1
w
(L0 − λ) w3 dy
R
gj (λ) ≡ Cj (λ) − F(λ) ,
F(λ) ≡
.
q
w dy
If Cj (0) > 1/2, then there exists an unstable real eigenvalue on
0 < λ < 3. Setting Cj (0) = 0, then gives
R 3
√
qU0 v0
w dy
1
R
.
1+
Dj2 =
3
3/2
2
K w dy
v0
Rigorous:
UBC – p. 44
Police Model I: Stability V
For q > 1, a K-hot-spot equilibrium on an interval of length S
L
, where
is unstable on an O(1) time-scale for any τu > 0 when D > DK
4
S
qU0
L
3
.
DK
≡ 2 2 2 4
1
+
ω
π
ω
8ε π α K 1 + cos K
Main Result:
Here ω is defined by
ω = S (γ − α) − U0 .
By a separate analysis involving the construction of asymmetric patterns,
the stability threshold with respect to the o(1) “small eigenvalues” is
4
qU
S
0
3
S
L
1
+
ω
DK
<
D
≡
.
K
16ε2 π 2 α2 K 4
ω
S
Notice that DK
is monotonically increasing in q. But, the optimal police
strategy is one that minimizes the stability threshold.
For a fixed U0 , it is not optimal for the police to be overly
focussed on drifting towards hot spots (observed numerically by L.
Ricketson).
Qualitative Result:
UBC – p. 45
Police Model I: A Hopf Bifurcation I
Consider two hot-spots, and let q = 3 so that the NLEP is solvable. Does
L
there exist a Hopf bifurcation when D < DK
whereby the amplitude of the
two hot-spots oscillate asynchronously on an O(1) time-scale?
For q = 3 and two-hot-spots, the function F(λ) in the NLEP problem
R q−1
−1
w
(L0 − λ) w3 dy
R
g1 (λ) ≡ Cj (λ) − F(λ) ,
F(λ) ≡
,
q
w dy
is F(λ) = 3/ [2(3 − λ)], and so λ is a root of
b
3χ01
λ=a+
, a≡3 1−
,
1 + τ̂ λ
2
Recall:
τ̂ ≡ εq−3 τu
R
q
w dy
q/2
v0
(simply set q = 3 and K = 2).
!
,
b≡−
9χ01 κp
.
2
√
U0 v0
κp ≡ R q
.
K w dy
UBC – p. 46
Police Model I: A Hopf Bifurcation II
For q = 3 and K = 2, there exists a Hopf Bifurcation at
H
L
with λ = ±iλI , when 0 < DK
< D < DK
. We have,
Main Result:
τu = τuH
3/2
τuH
v D13
,
= R0 3
a w dy
p
λI = −a(a + b) .
H
H
There is an explicit formula for DK
. No Hopf bif. if D < DK
.
L
H
, we predict asychronous
< D < DK
If τu > τuH on 0 < DK
oscillatory instability of the two spots. Numerically, the bifurcation looks
H
supercritical. If D < DK
, then we have stability for all τu .
Qualitative:
There is an intermediate range of police diffusivity (slower
than criminals) for which the two hot-spots exhibit asynchronous
oscillations. Maximum criminal activity is displaced periodically in time
from one hot-spot to its neighbour.
Interpretation:
Similar behavior for is expected for q 6= 3, but is more difficult to
analyze. (no explicit analytical formulas available).
Remark:
UBC – p. 47
Police Model II: Predator-Prey Interaction
On the basic cell |x| > l, and with q > 1, the
leading order asymptotics for A, P , and U , in the hot-spot region near
x = 0 is
Principal Result (Equilibrium):
1
U0
q
2
[w (x/ε)] .
A ∼ √ w (x/ε) , P ∼ [w (x/ε)] , U ∼
εb
ε v0
√
R q
Here b ≡ w dy, and w = w(y) = 2sech (y) is the homoclinic of
w′′ − w + w3 = 0. The amplitude of the hot-spot is determined by v0 , where
R 2+q
R 2+q
Z
w
dy
w
dy
1
2q
3
R
R
√
w dy = 2l(γ − α) − U0
,
=
.
q+1
v0
wq dy
wq dy
Remark I:
A hot-spot solution ceases to exist if
U0 ≥ U0c ≡ (q + 1)l(γ − α)/q.
For U0 < U0c , for a K-spot equilibrium on a domain of length S,
let l = S/(2K) and replace U0 → U0 /K. Then, use a glueing technique to
construct the multi-pulse pattern.
Remark II:
UBC – p. 48
Police Model II: Stability I
Let q > 1, and U0 < U0c so that a K-hot-spot equilibrium exists. Then:
Asymmetric hot-spot equilibria bifurcate from
the symmetric K-hot-spot branch at the threshold
2
3
2U0 q
2q
U
ω
S
0
S
> 0.
,
ω ≡ S(γ − α) −
1+
DK ≡
16ε2 π 2 α2 K 4
(q + 1)ω
q+1
Small Eigenvalue Threshold:
(Note that the amplitude of the hot-spot is proportional to ω.)
The NLEP now involves 3 separate nonlocal terms. When
τu ≪ O(εq−3 ), the stability threshold for this NLEP is
2
> DKS .
DKL = DKS
1 + cos (π/K)
NLEP Analysis:
By simple calculus we study DKS as a function of q for U0 fixed with
U0 < U0c . Optimal strategy is to minimize the threshold.
UBC – p. 49
Police Model II: Stability II
Key Qualitative Features (Optimal Strategy):
For q → ∞, then DKS → ∞, which gives a poor police strategy. Overly
focused aggressive police can lead (paradoxically) to rather closely
spaced stable hot-spots.
DKS has a minimum wrt q at some interior point in 1 < q < ∞. The
minimum value can be rather small only if U0 is relatively large. Thus,
only if there is enough police should they really focus their effort
towards hot-spots.
Example:
Curve U0
S = 1, γ = 2, α = 1, K = 2, ε = 0.02 Top Curve: U0 = 0.28, Bottom
= 0.44: Note: (U0c = 0.5).
0.9
0.8
0.7
S
DK
0.6
0.5
0.4
0.3
0.2
1.0
2.0
3.0
4.0
5.0
6.0
7.0
q
UBC – p. 50
Further Directions
For the basic urban crime model in 2-D with D = O(1):
Investigate effect of spatial heterogeneity of γ, α.
Derive the scalar nonlinear elliptic PDE, associated with a
saddle-node point, governing peak insertion.
Dynamics in 1-D and 2-D: Derive ODE’s for the locations of centers of a
collection of hot-spots (repulsive interactions?).
Are dynamic peak-insertion events for a collection of hot-spots
possible? If hot-spots become too closely spaced, we anticipate
annihilation. Can annihilation and creation events lead to
spatial-temporal chaos?
Analyze other models of the effect of police, incorporating dynamic
deterrence, i.e. A. Pilcher, EJAM (2010).
References:
T. Kolokolnikov, M.J. Ward, J. Wei, The Stability of Steady-State Hot-Spot
Patterns for a Reaction-Diffusion Model of Urban Crime, to appear
DCDS-B, (2012), (34 pages).
T. Kolokolnikov, S. Tse, M.J. Ward, J. Wei, Urban Crime, Hot-Spot
Patterns, and the Effect of Police: a Three-Component Reaction-Diffusion
UBC – p. 51
Model, in preparation, (2012).